How can I build a std::vector and then sort them?

You can just do

std::sort(data.begin(), data.end());

And it will sort your strings. Then go through them checking whether they are in order

if(names.empty())
    return true; // empty vector sorted correctly
for(std::vector<std::string>::iterator i=names.begin(), j=i+1; 
        j != names.end(); 
        ++i, ++j)
    if(*i > *j)
        return false;
return true; // sort verified

In particular, std::string::compare couldn’t be used as a comparator, because it doesn’t do what sort wants it to do: Return true if the first argument is less than the second, and return false otherwise. If you use sort like above, it will just use operator<, which will do exactly that (i.e std::string makes it return first.compare(second) < 0).